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Why control the temperature and pressure?

- isothermal and isobaric simulations (NPT) are most relevant to experimental data
- constant NPT ensemble: constant number of particles, pressure, and temperature

Causes of temperature and pressure fluctuations

the temperature and pressure of a system tends to drift due to several factors:

- drift as a result of integration errors
- drift during equilibration
- heating due to frictional forces
- heating due to external forces

Temperature coupling methods in GROMACS

weak coupling

- exponential relaxation Berendsen temperature coupling (Berendsen, 1984)

extended system coupling

- oscillatory relaxation Nosé-Hoover temperature coupling (Nosé, 1984; Hoover, 1985)

Berendsen temperature coupling

- there is weak coupling to an external ‘heat bath’
- deviation of system from a reference temperature To is corrected
- exponential decay of temperature deviation

the temperature of a system is related to its kinetic energy, therefore, the temperature can be easily altered by scaling the velocities vi by a factor λ

- is the temperature coupling time constant
- need to specify in input file (*.mdp file)

Some notes on Berendsen weak coupling algorithm

- very efficient for relaxing a system to the target temperature
- prolonged temperature differences of the separate components leads to a phenomenon called ‘hot-solvent, cold-solute’, even though the overall temperature is at the correct value

Solutions:

- apply temperature coupling separately to the solute and to the solvent problem with unequal distribution of energy between the different components

solutionscontinued …

- stochastic collisions (Anderson, 1980)

- a random particle’s velocity is reassigned by random selection from the Maxwell-Boltzmann distribution at set intervals does not generate a smooth trajectory, less realistic dynamics

- extended system (Nosé, 1984; Hoover 1985)

- the thermal reservoir is considered an integral part of the system and it is represented by an additional degree of freedom s

- used in GROMACS

Nosé-Hoover extended system

- canonical ensemble (NVT)
- more gentle than Anderson where particles suddenly gain new random velocities
- the Hamiltonian is extended by including a thermal reservoir term s and a friction parameter ξ, in the equations of motion

H = K + V + Ks + Vs

Nosé-Hoover extended system

- The particles’ equation of motion:
- ξ is a dynamic quantity with its own equation of motion:
- is proportional to the temperature coupling time constant (specified in *.mdp file)

the strength of coupling between the reservoir and the system is determined by

- when is too high slow energy flow between system and reservoir

- when is too low rapid temperature fluctuations

Nosé-Hoover produces an oscillatory relaxation, it takes several times longer to relax with Nosé-Hoover coupling than with weak coupling

- can use Berendsen weak coupling for equilibration to reach desired target, then switch to Nosé-Hoover
- Nosé-Hoover chain: the Nose-Hoover thermostat is coupled to another thermostat or a chain of thermostats and each are allowed to fluctuate

Pressure coupling

- The system can be coupled to a ‘pressure bath’ as in temperature coupling

weak coupling:

exponential relaxation Berendsen pressure coupling

extended ensemble coupling:

oscillatory relaxation Parrinello-Rahman pressure coupling (Parrinello and Rahman, 1980, 1981, 1982)

Berendsen pressure coupling

- equations of motion are modified with a

first order relaxation of P towards a reference Po

- rescaling the edges and the atomic coordinates ri at each step by a factor u leads to volume change
- u is proportional to β which is the isothermal compressibility of the system and which is the pressure coupling time constant. Both values must be specified in *.mdp file

Berendsen scaling can be done:

1. isotropically – scaling factor is equal for all three directions i.e. in water

2. semi-isotropically where the x/y directions are scaled independently from the z direction i.e. lipid bilayer

3. anisotropically – scaling factor is calculated independently for each of the three axes

Parrinello-Rahman pressure coupling

- volume and shape are allowed to fluctuate
- extra degree of freedom added, similar to Nosé-Hoover temperature coupling, the Hamiltonian is extended

box vectors and W-1 are functions of M

- W-1determines the strength of coupling

have to provide βand

in the input file (*.mdp file)

if your system is far from equilibrium, it may be best to use weak coupling (Berendsen) to reach target pressure and then switch to Parrinello-Rahman as in temperature coupling

- in most cases the Parrinello-Rahman barostat is combined with the Nosé-Hoover thermostat
- the extended methods are more difficult to program but safer

Weak coupling in *.mdp file

; OPTIONS FOR WEAK COUPLING ALGORITHMS =

; Temperature coupling =

tcoupl = berendsen

; Groups to couple separately =

tc-grps = Protein SOL_Na

; Time constant (ps) and reference temperature (K) =

tau-t = 0.1 0.1

ref-t = 300 300

; Pressure coupling

Pcoupl = berendsen

Pcoupltype = isotropic

; Time constant (ps), compressibility (1/bar) and reference P (bar) =

tau-p = 1.0

compressibility = 4.5E-5

ref-p = 1.0

Extended system coupling in *.mdp file

; OPTIONS FOR WEAK COUPLING ALGORITHMS =

; Temperature coupling =

tcoupl = nose-hoover

; Groups to couple separately =

tc-grps = PROTEIN SOL_Na

; Time constant (ps) and reference temperature (K) =

tau-t = 0.5 0.5

ref-t = 300 300

; Pressure coupling =

Pcoupl = parrinello-rahman

Pcoupltype = isotropic

; Time constant (ps), compressibility (1/bar) and reference P (bar) =

tau-p = 5.0

compressibility = 4.5E-5

ref-p = 1.0

References

- Berendsen, H.J.C., Postma, J.P.M., DiNola, A., Haak, J.R. Molecular dynamics with coupling to an external bath. J. Chem. Phys.81:3684-3690, 1984
- Nosé, S. A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys. 52:255-268, 1984
- Hoover, W.G. Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A31:1695-1697, 1985
- Berendsen, H.J.C. Transport properties computed by linear response through weak coupling to a bath. In: Computer Simulations in Material Science. Meyer, M., Pontikis, V. eds. Kluwer 1991, 139-155
- Parrinello, M., Rahman, A. Polymorphic transitions in single crystals: A new molecular dynamics method. J. Appl. Phys. 52:7182-7190, 1981
- Nosé, S., Klein, M.L. Constant pressure molecular dynamics for molecular systems. Mol. Phys. 50: 1055-1076, 1983

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